| The abc-conjecture for Algebraic Numbers |
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| 引用本文： | Jerzy BROWKIN. The abc-conjecture for Algebraic Numbers[J]. 数学学报(英文版), 2006, 22(1): 211-222. DOI: 10.1007/s10114-005-0624-3 |
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| 作者姓名： | Jerzy BROWKIN |
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| 作者单位： | Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL-02-097 Warsaw, Poland |
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| 摘 要： | The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.
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| 关 键 词： | abc猜想 非零整数 代数数字 Fermat理论 |
| 收稿时间： | 2003-10-23 |
| 修稿时间： | 2003-10-232004-06-18 |
The abc–conjecture for Algebraic Numbers |
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| Jerzy Browkin. The abc–conjecture for Algebraic Numbers[J]. Acta Mathematica Sinica(English Series), 2006, 22(1): 211-222. DOI: 10.1007/s10114-005-0624-3 |
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| Authors: | Jerzy Browkin |
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| Affiliation: | (1) Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL–02–097 Warsaw, Poland |
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| Abstract: | The abc–conjecture for the ring of integers states that, for every ε > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + ε with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m. In the present paper we propose an abc–conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number. From this abc–conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over ℚ. |
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| Keywords: | abc– conjecture Radical Height Fermat's last theorem for algebraic numbers |
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